Snatch loading of a single taut moored floating wave energy converter due to focussed wave groups

determining the response of a sandy beach in the lee of the farm, and should be accordingly heeded in applying a wave farm for coastal defence

To accomplish this objective, four scenarios are compared, corresponding to three locations of the wave farm at different distances from the coast, plus the baseline (no farm) scenario, under different wave conditions.First, the impacts of the wave farm on the wave conditions are examined using a nearshore wave propagation model, SWAN (Booij et al., 1996).This is a phase-averaged spectral model that computes the effects of the wave farm using an energy transmission coefficient, whose values are obtained from the laboratory tests carried out by Fernandez et al. (2012).The wave farm is implemented on a high-resolution grid at different distances from a reference (10 m water depth) contour: (i) 2 km, (ii) 4 km; and (iii) 6 km.Second, based on the results of the aforementioned scenarios a coastal processes model, XBeach (Roelvink et al., 2006), is used to compare the effects of the wave farm at the different locations with the baseline scenario.A set of impact indicators is developed, specifically, to quantify these effects and establish the role played by the farm-to-coast distance.This methodology is applied to a case study at Perranporth Beach, Cornwall (UK).
A 3.5 km long sandy beach facing directly toward the North Atlantic Ocean, Perranporth is in an area with a great potential for marine renewable energy (Thorpe, 2001) -as corroborated by the Wave Hub pilot test site.The extremely energetic storms of February 2014 proved that Perranporth is subject to increased erosion risks from rising sea level and storminess (CISCAG, 2011).In view of these risks, and given that a wave farm consisting of floating WECs adapts naturally to any sea level changes (Martinelli et al., 2008), Perranporth constitutes a prime area for using such wave farms for coastal protection.

WAVE PROPAGATION MODEL
The wave propagation was computed using SWAN (Simulating WAves Nearshore), a third-generation spectral wave model based on the action balance equation (Holthuijsen, 2007): in which the first term on the left-hand side of this equation represents the variation of wave action density (N) in time (t), the second and third term the velocity propagation in the geographical space (with c x and c y the propagation velocity in x-and y-space, respectively) and the fourth and fifth term the propagation velocity in the θand σ-space, respectively (where θ represents the direction and σ the relative frequency).
On the right-hand side of Equation ( 1), the source term S represents the effects of generation, dissipation and nonlinear wave-wave interactions.Two computational grids with different resolutions were defined (Figure 2): (i) an offshore grid covering an area of approx.100 km × 50 km with a grid size of 400 × 200 m, and (ii) a high-resolution nearshore (nested) grid covering the study area, with dimensions of approx.8 km × 6 km and a grid size of 16 m × 12 m.Onto the latter grid was implemented the wave farm, which consisted of 11 WaveCat WECs (Fernandez et al., 2012;Iglesias et al., 2008) arranged in two rows, with a spacing between devices of 2.2D, where D = 90 m is the distance between the twin bows of a single device (Carballo and Iglesias, 2013).Two wave conditions (Table 1) were prescribed at the outer (ocean) boundary of the offshore grid based on the analysis of the offshore wave climate in the area (Kenney, 2009).Wave transmission through the wave farm was modelled following the same approach as in a previous successful application of the model (Abanades et al., 2014).For the purposes of the sensitivity analysis three locations of the wave farm were considered, at distances of 2 km, 4 km and 6 km from the reference (10 m water depth) contour -corresponding to water depths of approx.25 m, 30 m and 35 m, respectively (Figure 3).To measure the impact of the wave farm on the wave conditions in its lee an impact indicator was defined: the Reduction in the Significant wave Height, where the subindex i refers to the position of the wave farm, and H s,b and H s,fi are the significant wave height in the baseline scenario and with the wave farm, respectively, at a point of the coast designated by its coordinates (x,y), with the x-coordinate referring to the easting and the y-coordinate to the northing.This nondimensional indicator quantifies the shadow caused by the wave farm in its lee.
The performance of the wave farm at the different positions was also analysed by means of the wave power (J, in units of Wm -1 in the SI), which is computed in SWAN from its x-and y-components: ( ) where ρ is the water density, g is the acceleration due to gravity, and E(σ,θ) is the directional spectral density, which specifies how the energy is distributed over frequencies (σ) and directions (θ).The wave power magnitude is then given by ( )

COASTAL PROCESSES MODEL
Based on the results of the wave propagation model, the coastal processes model, XBeach, was used to compute the impact of the wave farm on beach morpholology.
XBeach is a 2DH (two-dimensional horizontal) time-dependent model that solves coupled cross-shore and alongshore equations for wave propagation, flow, sediment transport and bottom changes.The full description of the model can be found in Roelvink et al. (2006).
The sediment transport module solves the depth-averaged advection diffusion equation (Galappatti and Vreugdenhil, 1985) on the time scale of wave groups (Baldock et al., 2011), where C is the wave group varying depth averaged, sediment concentration, D h is the sediment diffusion coefficient, which is represented by an adaptation time, T s , that is based on the local water depth, h, and sediment fall velocity.The terms u E and v E represents the depth-averaged velocities and C eq the equilibrium concentration, representing the source term in the right hand side of the equation.The sediment transport formula defined by Van Thiel de Vries ( 2009) was chosen to determine the sediment equilibrium concentration.
XBeach has been widely validated to determine the impact of storms on sandy (McCall et al., 2010;Pender and Karunarathna, 2013;Roelvink et al., 2009) and gravel beaches (Jamal et al., 2014;McCall et al., 2013;McCall et al., 2012;Williams et al., 2012)   The following parameters obtained from the wave propagation model along the line A-A' (Figure 5) were the input of XBeach for the analysis of the beach response under the storm events: root mean square wave height (H rms ), mean absolute wave period (T m01 ), mean wave direction (θ m ) and directional spreading coefficient (s).As for tidal effects, Perranporth is a macrotidal beach with a Mean Spring tidal Range, MSR, of 6.3 m (Austin et al., 2010), and consequently their influence on the response of the beach ought to be considered (Masselink and Short, 1993).The MSR was included in the model with a semi-diurnal tidal regime (two low and two high tides each day).
The effects of the wave farm on the beach morphology were determined based on a comparison of the different wave farm scenarios with the baseline (no farm) case.The following impact indicators were defined: (i) bed level impact (BLI i ), (ii) beach face eroded area (FEA b or FEA i ), (iii) non-dimensional erosion reduction (NER i ), and (iv) mean cumulative eroded area (CEA b or CEA i ).The indicators corresponding to the baseline and wave farm scenarios were denoted with the subscripts b or i, respectively, with i indicating the farm-to-coast distance (i = 2km, 4km or 6km).
The bed level impact (BLI i ), with units of m in the S.I., was defined as the change in bed level in the i-th scenario.A positive value signifies that the seabed level is higher in the presence of the wave farm.
The beach face is the area over the mean water level exposed to the action of the waves.In order to quantify the wave farm effects on this particularly relevant area of the beach, the beach face eroded area (FEA), with units of m 2 in the S.I., was defined in the wave farm (FEA f,i ) and baseline (FEA b ) scenarios by where ζ 0 (x,y) is the initial bed level at the point of coordinates (x,y), and x 1 and x max are the values of the x-coordinate at the seaward end of the beach face and landward end of the profile, respectively.These indicators are profile functions (the profile being designated by the y-coordinate) that evaluate the erosion caused by the storms at the beach face, thus enabling to compare the reduction of the erosion obtained in each of the three wave farm scenarios relative to the baseline case.
The non-dimensional erosion reduction, is also a profile function, in this case non-dimensional, which computes the variation in the eroded area of a generic profile (y) as a fraction of the total eroded area of the same profile brought about by the wave farm.A positive or negative value implies a reduction or increase in the eroded area, respectively, as a result of the wave farm.
Finally, the mean cumulative eroded area (CEA), with units of m 2 (or m 3 per linear metre of beach), was determined as well in the baseline scenario (CEA b ) and with the different wave farms (CEA f,i ).For its definition three reference profiles were considered: P A , P B and P C (Figure 2).For each of these the beach was divided into two parts, to the north ( where the integration variables χ and y correspond to integration along the profile and the beach, respectively.The integration limits along the profile are: x 0 , the value of the x-coordinate corresponding to the first point of the profile (seaward end); x max , the (landward) end of the profile.Along the beach, the integration limits are: y 0 and y max , the values of the y-coordinate corresponding to the southernmost and northernmost points of the beach, and y P the value corresponding to the reference profile.The CEA indicator represents the average cumulative eroded area of the two sections of the beach along the profile (x).A positive value signifies that the mean volume of material along the section of the beach is reduced compared with the initial situation (erosion).

RESULTS
First, the results obtained from the nearshore wave propagation model were analysed to study the impact of the wave farm on the wave conditions.The nearshore significant wave height (H s ) for the different scenarios (baseline and with the wave farm at distances of 2 km, 4 km and 6 km from the reference contour) is shown in Figure 5 for CS1 (Table 1).The reduction in the significant wave height in the lee of the farm caused by the energy extraction is apparent.This reduction was assessed by means of the impact indicator RSH i (Figure 6) defined in Section 3.1.The maximum value of the indicator was achieved within the second row of WECs with values of up to 50%.At a distance of 1.5 km from the second row of devices, the reduction reached a peak of 40% due to the merging of the shadows caused by the first and the second row of devices.
However, this reduction decreased moving towards the coast due to the redistribution of the energy from the edges into the shadow caused by the wave farm.At a water depth of 10 m, the average reduction caused by the wave farm closest to the coast (2 km) was approx.25%, whereas for the wave farm at 4 and 6 km the average reduction was approx.15% and 9%, respectively.The relevance of the farm-to-coast distance may be readily observed in the shadows caused by the wave farm at different distances.The area affected at the coastline by the wave farm furthest to the coast (6 km) was greater than 7 km, however the average reduction of the significant wave height in this area was less than 5%.On the other hand, the wave farm at 2 km affected a smaller area in the coastline, around 4 km, but the reduction exceeded 10%. Figure 7 shows this reduction for CS1 (above) and CS2 (below) along the line AA´, located in Figure 5, which corresponded to the area of interest at Perranporth Beach and was used as input for the coastal processes model.
This figure confirmed the different shadow pattern brought about by the wave farm at a distance of 4 and 6 km compared with the 2 km.In the latter, the reduction mainly occurred in the central section of the beach, being less significant in the northern area of the beach.However, for the other two scenarios, the reduction was found to be approx.constant along the line AA´.In terms of wave power, the resource was evaluated using equations ( 2), ( 3) and ( 4) at the location of each of the WECs in the wave farm.Table 2 shows the overall wave power incident on the wave farm for the different distances, it was found that the closer the wave farm to the coast, the lesser the resource, due to the dissipation caused by the different coastal processes that occur in intermediate and shallow water.For the wave farm closest to the coast the reduction of the wave power compared to the scenario with the wave farm at a distance of 6 km was 10.5% and 8.7% for CS1 and CS2, respectively.In the case of the wave farm at 4 km the reduction compared with the scenario at 6 km is 5.7% and 7.3% for CS1 and CS2, respectively.In summary, on the one hand the wave farm closest to the beach caused the greatest reduction in the significant wave height, but, on the other hand, the resource in that area is lower than in deeper areas, and, therefore, a comparative study of the response of the beach under storm conditions is necessary to determine the best location for a wave farm in terms of wave energy resource and coastal protection.
Case  Second, the coastal processes model used the output of the wave propagation model to study in which manner the modification of the wave patterns affected the coastal processes and, consequently, the beach morphology.To quantify this alteration the results were analysed by means of the impact indicators defined in Section 3.2.The first indicator was the bed level difference, BLI, which represented the difference of the bed level between the baseline and the wave farm scenarios at a point in time.Figure 8 shows BLI values at the end of the storm for CS1 with the wave farm at a distance of 2 km (left), 4 km (middle) and 6 km (right).It was observed that the main impact caused by the wave farm was located at the beach face, where reductions of the erosion up to 1.5 m were found.Figure 9 illustrates the evolution of three profiles at the end of the storm for CS2, corresponding with three section of the beach that were identified by their different responses under the storms: (i) the southern section of the beach (P1 in Figure 9) with a smooth slope in the intertidal area; (ii) the area backed by a very steep dune (P2 in Figure 9) where the mean water level was close to the toe of the dune; and (iii) the northern section of the beach (P 3 in Figure 9) also backed by the dune, but with a greater distance from the toe of the dune to the mean water level.In the case of P 1 and P 3 the main erosion occurred on the beach face and this material was moved to lower sections of the beach, however in the case of P2 accretion was detected in the intertidal area due to the material eroded in the steep dune for the proximity of the mean water level to it.
In the area of P3 were found the greatest values of the BLI indicator, with reductions greater than 1 m, while in the section P1 the reduction took values of approx.0.5 m.In the comparison between scenarios, the wave farm at a distance of 2 km caused greater reduction of the erosion in the beach along the beach than the other scenarios, in which areas with significant reductions of erosion were combined with negligible values or even accretion.In the lower sections of the beach, accretion occurred due to the amount of material eroded in the beach face.In the scenario with the wave farm closest to the coast the BLI took negative values of -0.5 m in the southern area of the beach, which meant that the accretion without the farm was bigger than with it, due to the greater erosion produced in the intertidal area in the baseline scenario.This reduction of the accretion with the wave farm at a distance of 4 km and 6 km took place only in a few sections of the beach with BLI values less than 0.3 m.
The impact factor FEA was defined to quantify the erosion in the beach face along the beach (Figure 10).The greatest values of this indicator along the beach were focussed in the southern area because this section was not backed by the dune.The erosion in the baseline scenario was, in general, greater than the scenarios with the wave farm, especially in the middle and northern area of the beach, y-coordinate (along the beach) > 1250 m.To compare the reduction between the different wave farm scenarios the indicator NER was defined, which showed the variation of the erosion in terms of the eroded area in the baseline scenario (Figure 11).The NER values fluctuated considerably along the beach, but it was observed that the reduction using a wave farm at a distance of 2 km was greater than the other two scenarios.In the area of the steep dune (500 m < y-coordinate < 1250 m), the erosion in the beach face was very low (negligible in some sections), and very few profiles presented an isolated response taking the NER factor negative values (greater erosion with the farm than without it).However, in terms of the average reduction of the beach face erosion along the whole beach, it was confirmed that the wave farm at 2 km offered a greater degree of coastal protection, around 15% in both case studies, than the scenario with the wave farm at 4 and 6 km, which presented an approximate reduction of approx.
10%.Considering particular sections of the beach, the impact was much more significant, for instance, the reduction exceeded 20% for the wave farm at 2 km for values of the y -coordinate between 1200 and 2000 m in CS2, which was the area most affected by the reduction of the significant wave height (Figure 7).The results for the wave farm at 4 and 6 km did not present large differences in terms of the reduction of the erosion along the whole beach; however the average reduction for the farm at 4 km was slightly greater (13%) than the farm at 6 km (11%) in the area backed by the dune (y > 1250 m).Finally, the CEA indicator computed the volume of material moved per linear metre along the beach between the initial conditions and the last point of the simulation for the different scenarios.This indicator was applied to the northern (CEA N ) and southern (CEA S ) section of the beach taking as reference for each case the following profiles (Figure 4): P A (south), P B (middle) and P C (north), which allowed the variations in the longshore sediment transport to be studied.Figure 12 shows the evolution of this factor along the profile (x) for CS1, where the negatives values represented an increase in the volume of material with respect to the initial conditions (accretion).In the lowest section of the profile, the volume of material for the scenarios studied was larger than the initial volume due to the material eroded, mainly from the following sections along the profile: (i) the beach face (1200 m < x-coordinate < 1300 m) and (ii) the area that faced the storms in low tide (800 m < x-coordinate < 1000 m), which was more significant in the southern area of the reference profiles.The geomorphological complexity of the southern section of the beach resulted in very different behaviour between the different scenarios.The wave farm at a distance of 2 km presented a significant rise in the volume of material in the southern area of the beach, especially taking as reference the profiles P A and P B .This could be associated to the modification of the wave patterns brought about by the wave farm, given that the main reduction of the significant wave height occurred in the southern and middle area of the beach.Therefore, part of the material eroded in the northern section, where the reduction of the significant wave height was less, could be moved to the southern part of the beach, increasing the volume in this section.As for the wave farm at 4 and 6 km, they did not present significant differences compared with the baseline scenario, nonetheless the erosion caused in the absence of the farm was greater.For instance, in the northern area of the different profiles, it is observed that the greatest accretion at x = 1250 m occurred in the baseline scenario due to the largest amount of material eroded at the beach face.In the case of the profile P A , this was followed by the scenario with the wave farm at a distance of 2 km, associated with the material moved from the north of the beach, but for the profiles P B and P C , the greatest values of accretion, after the baseline scenario, occurred with the wave farm at 4 and 6 km given that the farm at these distances reduced the erosion less than the scenario at 2 km.To sum up, a wave farm can alter not just the wave conditions in its lee but also the morphology and the sediment transport of the beach.

CONCLUSIONS
The role played by the distance in the impact of a wave farm on the beach morphology was analysed in this paper.To investigate this, a wave farm formed by 11 WECs was located at different distances: 2 km, 4 km and 6 km from a reference (10 m water depth) contour in a high-resolution suite of numerical models.This suite consisted of a nearshore wave propagation model coupled to a coastal processes model, which allowed the impacts of the wave farm on wave conditions and coastal processes to be assessed.
The wave farm extracted energy from the waves, which was characterised by means of wave transmission coefficients that were obtained in laboratory tests.The comparison between the baseline and the wave farm scenarios showed the importance of the farm-to-coast distance, given that, depending on the location, the area affected by the farm and the magnitude of the wave height reduction varied considerably.In the case of the wave farm at a distance of 6 km, the impact of the farm covered an area of 7 km along the coast but the reduction of the significant wave height at a water depth of 10 m was less than 10%; nonetheless, the area affected by the wave farm closest to the farm was 4 km and the reduction approx.25% due to less energy being diffracted into the shadow of the farm situated closer to the beach.
The impact of the wave farm on the wave conditions resulted in an alteration of the coastal processes nearshore, and therefore, of the beach morphology.To quantify this, a suite of impact indicators was developed and applied to the results of the different scenarios.The reduction of the erosion brought about by the different wave farms was mainly in two areas of the beach: (i) the area at a water depth of approx.3 m, which faced the storms during the low tide; and (ii) the beach face of the beach.Whereas in the former, the reduction did not exceed 0.5 m, in the latter it reached 1.5 m.In the comparison between scenarios, the wave farm at 2 km offered greater reductions of the erosion than the farm at 4 and 6 km, which presented similar responses.The overall reduction of the erosion on the beach face compared to the baseline scenario was 15% for the closest wave farm and approx.10% for the other two.These values fluctuated significantly along the beach, and in some sections, especially in the northern area of the beach, exceeded 40%.
Lastly, it was also found that the wave farm may change the distribution of sediment along the beach.The alteration of the wave conditions with the farm at 2 km modified the sediment transport patterns, increasing the volume of material moved to the southern area of the beach.This confirmed that the effects on the beach morphology of the wave farm closest to the coast were more pronounced than in the other scenarios; nevertheless, the overall wave resource in this area was 10 % less than with the case with the furthest farm (6 km) due to the attenuation of wave energy caused by the coastal processes that occur in shallow waters.When comparing the scenarios at 4 km and 6 km, the impact on the beach morphology did not present significant differences but there were differences in the overall resource at the wave farm: 8% less at 4 km than at 6 km.
In summary, the selection of the location for a wave farm is not trivial.In this work, it has been proven that the degree of coastal protection afforded by a wave farm varies significantly as a function of its distance to the coastline.Thus, after this study, the effects of the wave farm on the coast may become one of the main considerations in this selection, not least in areas subject to erosion risks, where the wave farms can contribute considerably to coastal defence.

Figure 1 :
Figure 1: Bathymetry of SW England [water depths in m] including the location of Perranporth Beach, Wave Hub Project and an aerial photo of Perranporth Beach [source: Coastal Channel Observatory].

Figure 2 :
Figure 2: Computational grids of the wave propagation (SWAN) and coastal processes (XBeach) model [water depths in m]

Figure 3 :
Figure 3: The three locations considered for the wave farm, at distances of 2 km, 4 km and 6 km from the reference (10 m water depth) contour [water depth in m].
at different locations.In this case, the impact of the wave farm on the beach morphology (3D) was compared to the baseline scenario at Perranporth Beach following the model set up applied byAbanades et al. (2014) at the same location to study the evolution of the beach profile (2D).The high-resolution grid implemented on XBeach covered an area of 1.4 km cross-shore and 3.0 km alongshore at Perranporth Beach with a resolution of 6 m and 12.5 m, respectively.The bathymetry data, from the Coastal Channel Observatory, were interpolated onto this grid (Figure4), which comprised elevation values from -20 m to more than 30 m with reference to the local chart datum (LCD).

Figure 4 :
Figure 4: Bathymetry of Perranporth Beach computed in XBeach.Profiles P1, P2 and P3 included.Water depth in relation to local chart datum [in m].
ζ f,i (x,y) and ζ b (x,y) are the seabed level in the wave farm and baseline scenarios, respectively, at a generic point of the beach designated by its coordinates (x,y) in the horizontal reference plane.The y-coordinate axis follows the general orientation of the beach, with values increasing towards the northern end of the beach, and the x-coordinate is the horizontal coordinate along the profiles, with values increasing towards the landward end of the profile.Thus, the BLI i indicator represents

Figure 5 :
Figure 5: Significant wave height [m] in the baseline scenario and in the presence of the farm at distances of 2 km, 4 km and 6 km from the reference (10 m water depth) contour in CS1 (clockwise from above left).

Figure 6 :
Figure 6: Reduction of the significant wave height (%) with the wave farm at a distance of: 2 km (RSH 2km ), 4 km (RSH 4km ) and 6 km (RSH 6km ) from the reference (10 m water depth) contour in study CS2 [in m].

Figure 7 :
Figure 7: Significant wave height [in m] in the baseline scenario (Hs, b) and in the presence of the farm at a distance of: 2 km (Hs,2 km), 4 km (Hs,4 km) and 6 km (Hs,6 km) from the reference (10 m water depth) contour across the line AA' in CS1 (above) and CS2 (below).

Figure 8 :
Figure 8: Bed level impact with the wave farm at a distance of 2 km (BLI 2km ), 4 km (BLI 4km ) and 6 km (BLI 6km ) at the end of the storm in CS1.

Figure 9 :
Figure 9: Bed level at Profiles P 1 , P 2 and P 3 : initial (ζ 0 ) and at the end of the storm in CS2 in the baseline scenario (ζ b ) and with the wave farm at a distance of 2 km (ζ f2km ), 4 km (ζ f4km ) and 6 km (ζ f6km ).

Figure 10 :
Figure 10: Beach face eroded area in the following scenarios: baseline (FEA b ) and with the wave farm at a distance of 2 km (FEA 2km ), 4 km (FEA 4km ) and 6 km (FEA 6km ) along Perranporth Beach (y -coordinate, with y increasing towards the north of the beach) at the end of the storm in CS1 (above) and CS2 (below).

Figure 11 :
Figure 11: Non-dimensional erosion reduction (NER) at the beach face in the following scenarios: with the wave farm at a distance of 2 km (NER 2km ), 4 km (NER 4km ) and 6 km (NER 6km ) along Perranporth Beach (ycoordinate, with y increasing towards the north of the beach) at the end of the storm in CS1 (above) and CS2 (below).

Figure 12 :
Figure 12: Mean cumulative eroded area in the baseline scenario (CEA b ) and in presence of the wave farm at a distance of 2 km (CEA 2km ), 4 km (CEA 4km ) and 6km (CEA 6km ) from the reference (10 m water depth) contour in the northern area (above) and southern (below) across each of the reference profiles P A , P B and P C (Figure 4) at the end of the storm in CS1.The x-coordinate represents the distance along the profile, with x = 0 the most offshore point.